0321_1032_1302
Counting sequence:
1, 1, 2, 6, 21, 77, 289, 1107, 4322, 17162, 69137, 281917, 1161404, 4826652, 20211146, 85192214, 361185371, 1539206765, 6589630969, 28328237571, 122235870619, 529235187277, 2298474652128, 10010555479062, 43712371026957, 191333144596753, 839339285847866, 3689578089846098, 16249697898564755, 71694730329569823, 316848723240147607, 1402470353691880813, 6216856622722035264, 27595965215562590444, 122654454580835230417, 545823712943448760471, 2431776982828209480418, 10846025243557697272604, 48424855575596534540935, 216418366751397708641057, 968114760242338933622661, 4334579014926506072199227, 19423887496511924376015584, 87111918025389272576859896, 390980751294997403403240707, 1756122403605498927929959211, 7893361173762590739453239405, 35502914197965730341880188773, 159789676525933355376748945850, 719622453528811749011805231200, 3242807963112127210321285692770, 14621373845554260490875259085228, 65962335385480925287820821810502, 297739021190236221040399892922416, 1344617858714567558917430091219271, 6075434372160026168085348708464037, 27463982505149387161337120683521118, 124208178491254413299628389255675254, 561992499839778617333193632125449255, 2543888005084459290724050415229229447, 11519839080305585352655442190093991764, 52187910867673561456962869263678415728, 236517285279354550630708107732370610080, 1072309114176994217402555546045551163128, 4863357097845788949722505269501640962553, 22065131325690647490654335668795202107039, 100144345610307443351793615201371391437321, 454664952724587258457752542817251103825923, 2064892356590722825974860695554172076155969, 9380806857007387434300357747943090136435091, 42630058658449809646996759321722005209903322, 193785334191687328811944212080810786076570774, 881153474202882839471009121495426045166475141, 4007785467303369516907047166663517605729750251, 18233764846614810954930578520445831451334237495, 82978209072640253179182276813968147462263816003, 377715286394551029042144132911413952963436341509, 1719788148646152881752638569473340335955670005109, 7832356186331510511610961180302775728762543606161, 35679133641692048863123682829950472661547647221199, 162569088812817948841097730260053231983324171882023, 740902303830115379298648011392046438543455486324193, 3377386898410578515873338044709613038629240945159152, 15399097130945528756482171406861880793536027562394688, 70226664094548791637295313709740670030592786641580993, 320331020165049123924785428088964025783571119315818469, 1461450287022905865591758698653936024264346054459935495, 6668914905554146187693618546608390123038504684727643141, 30437602341476585628128365718470845816283107214349669212, 138946610090412354641134510747725775212279365550990290280, 634403994608178978025493040311046673225968589783568141277, 2897093692018815063504643362131561586022483245383990733093, 13232324707566272638734219721149918855987243250515112131338, 60448438872664647409923658105806171799832042548969542651942, 276189882933454639328128443789710315264636217752660159396638, 1262125772830316375933981477846676377596069118534044254243240, 5768570450229113136175351083244578131813002316926572915502803, 26369567883655148365415532932012713356471731061439923810257547, 120560673580499270909443052564045258032694283541173041730486810, 551283247601036265014078449451119701096277081839684690949968032, 2521210336702384642553816444000316324317181977970903674011659025
Implicit equation for the generating function in Maple syntax:
(x-1)*x^2*F(x)^3+x*(2*x^2-3*x+2)*F(x)^2+(x-1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) x^{2} F \! \left(x \right)^{3}+x \left(2 x^{2}-3 x +2\right) F \! \left(x \right)^{2}+\left(x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 289
a(7) = 1107
a(8) = 4322
a(9) = 17162
a(10) = 69137
a(n+11) = 8*n*(2*n+1)/(n+12)/(n+11)*a(n)-1/2*(293*n^2+754*n+456)/(n+12)/(n+11)*a(n+1)+1/4*(2359*n^2+10829*n+12132)/(n+12)/(n+11)*a(n+2)-1/4*(5579*n^2+36901*n+60192)/(n+12)/(n+11)*a(n+3)+3/8*(5797*n^2+50283*n+108048)/(n+12)/(n+11)*a(n+4)-1/8*(18922*n^2+203843*n+545328)/(n+12)/(n+11)*a(n+5)+1/8*(14753*n^2+190168*n+609708)/(n+12)/(n+11)*a(n+6)-1/4*(4123*n^2+61820*n+230826)/(n+12)/(n+11)*a(n+7)+1/2*(805*n^2+13733*n+58395)/(n+12)/(n+11)*a(n+8)-3/4*(n+10)*(138*n+1253)/(n+12)/(n+11)*a(n+9)+1/2*(31*n+312)/(n+12)*a(n+10), n >= 11
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 289
a \! \left(7\right) = 1107
a \! \left(8\right) = 4322
a \! \left(9\right) = 17162
a \! \left(10\right) = 69137
a \! \left(n +11\right) = \frac{8 n \left(2 n +1\right) a \! \left(n \right)}{\left(n +12\right) \left(n +11\right)}-\frac{\left(293 n^{2}+754 n +456\right) a \! \left(n +1\right)}{2 \left(n +12\right) \left(n +11\right)}+\frac{\left(2359 n^{2}+10829 n +12132\right) a \! \left(n +2\right)}{4 \left(n +12\right) \left(n +11\right)}-\frac{\left(5579 n^{2}+36901 n +60192\right) a \! \left(n +3\right)}{4 \left(n +12\right) \left(n +11\right)}+\frac{3 \left(5797 n^{2}+50283 n +108048\right) a \! \left(n +4\right)}{8 \left(n +12\right) \left(n +11\right)}-\frac{\left(18922 n^{2}+203843 n +545328\right) a \! \left(n +5\right)}{8 \left(n +12\right) \left(n +11\right)}+\frac{\left(14753 n^{2}+190168 n +609708\right) a \! \left(n +6\right)}{8 \left(n +12\right) \left(n +11\right)}-\frac{\left(4123 n^{2}+61820 n +230826\right) a \! \left(n +7\right)}{4 \left(n +12\right) \left(n +11\right)}+\frac{\left(805 n^{2}+13733 n +58395\right) a \! \left(n +8\right)}{2 \left(n +12\right) \left(n +11\right)}-\frac{3 \left(n +10\right) \left(138 n +1253\right) a \! \left(n +9\right)}{4 \left(n +12\right) \left(n +11\right)}+\frac{\left(31 n +312\right) a \! \left(n +10\right)}{2 n +24}, \quad n \geq 11
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/379/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]*F[8,x]
F[3,x] = F[1,x]+F[4,x]+F[9,x]
F[4,x] = F[5,x]*F[8,x]
F[5,x] = F[3,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[0,x]*F[5,x]*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]*F[8,x]
F[10,x] = F[11,x]+F[3,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[17,x]*F[8,x]
F[13,x] = F[0,x]+F[14,x]+F[15,x]
F[14,x] = F[13,x]*F[8,x]
F[15,x] = F[16,x]
F[16,x] = F[0,x]*F[13,x]*F[8,x]
F[17,x] = F[1,x]+F[18,x]+F[4,x]
F[18,x] = F[17,x]*F[8,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right) F_{8}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{0}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{3}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{0}\! \left(x \right)+F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{0}\! \left(x \right) F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x)*F_8(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_9(x))
Eq(F_4(x), F_5(x)*F_8(x))
Eq(F_5(x), F_3(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_0(x)*F_5(x)*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x)*F_8(x))
Eq(F_10(x), F_11(x) + F_3(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_17(x)*F_8(x))
Eq(F_13(x), F_0(x) + F_14(x) + F_15(x))
Eq(F_14(x), F_13(x)*F_8(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_0(x)*F_13(x)*F_8(x))
Eq(F_17(x), F_1(x) + F_18(x) + F_4(x))
Eq(F_18(x), F_17(x)*F_8(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "row_and_col_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0], "pos": [[3, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[2, 1], [3, 1]]}, {"patt": [1, 0], "pos": [[3, 1], [3, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [3, 1]]}, {"patt": [0, 2, 1], "pos": 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